NUMBER SENSE AT A FLIP

1
NUMBER SENSE AT A FLIP
2
Number Sense

• Number Sense is memorization and practice.
  The secret to getting good at number sense is to
  learn how to recognize and then do the rules
  accurately . Then learn how to do them quickly.
  Every practice should be under a time limit.

3
The First Step
The first step in learning number sense should be
to memorize the PERFECT SQUARES from 12 1 to
402 1600 and the PERFECT CUBES from 13 1 to
253 15625. These squares and cubes should be
learned in both directions. ie. 172 289
and the 289 is 17.
4
2x2 Foil (LIOF)23 x 12
The Rainbow Method
Work Backwards
Used when you forget a rule about 2x2
multiplication

• The last number is the units digit of the
  
  product of the units
  digits
• Multiply the outside, multiply the inside
• Add the outside and the inside together plus any
  
  carry and write down the units digit
• Multiply the first digits together and add
  
  and carry. Write down the number

3(2)
2(2)3(1)
2(1)
6
7
2
276
5
Consecutive Decades35 x 45

• First two digits the small tens digit times
  one more than the large tens digit.
• Last two digits are always 75
• 3(41) 75
• 15 75

6
Ending in 5Tens Digits Both Even45 x 85

• First two digits the product of the tens
  digits plus ½ the sum of the tens digits.
• Last two digits are always 25
• 4(8) ½ (48) 25
• 38 25

7
Ending in 5Tens Digits Both Odd35 x 75

• First two digits the product of the tens
  digits plus ½ the sum of the tens digits.
• Last two digits are always 25
• 3(7) ½ (37) 25
• 26 25

8
Ending in 5Tens Digits OddEven35 x 85

• First two digits the product of the tens
  digits plus ½ the sum of the tens digits.
  Always drop the remainder.
• Last two digits are always 75
• 3(8) ½ (38) 75
• 29 25

9
Squaring Numbers Ending In 5752

• First two digits the tens digit times one more
  than the tens digit.
• Last two digits are always 25
• 7(71) 25
• 56 25

10
Multiplying By 12 ½ 32 x 12 ½
(1/8 rule)

• Divide the non-12 ½ number by 8.
• Add two zeroes.

32
400

8
4 00
11
Multiplying By 16 2/3 42 x 16 2/3
(1/6 rule)

• Divide the non-16 2/3 number by 6.
• Add two zeroes.

42
700

6
7 00
12
Multiplying By 33 1/3 24 x 33 1/3
(1/3 rule)

• Divide the non-33 1/3 number by 3.
• Add two zeroes.

• 24

800

3
8 00
13
Multiplying By 2532 x 25
(1/4 rule)

• Divide the non-25 number by 4.
• Add two zeroes.

32
8
00

4
8 00
14
Multiplying By 5032 x 50
(1/2 rule)

• Divide the non-50 number by 2.
• Add two zeroes.

32
16

00
2
16 00
15
Multiplying By 7532 x 75
(3/4 rule)

• Divide the non-75 number by 4.
• Multiply by 3.
• Add two zeroes.

32

8x32400
4
24 00
16
Multiplying By 12532 x 125
(1/8 rule)

• Divide the non-125 number by 8.
• Add three zeroes.
• 32

4000

8
4 000
17
Multiplying When Tens Digits Are Equal And
The Unit Digits Add To 1032 x 38

• First two digits are the tens digit
  times one more than the tens digit
• Last two digits are the product
  of the units digits.

3(31)
2(8)
12 16
18
Multiplying When Tens Digits Add To 10 And
The Units Digits Are Equal67 x 47

• First two digits are the product of the tens
  digit plus the units digit
• Last two digits are the product
  of the units digits.

6(4)7
7(7)
31 49
19
Multiplying Two Numbers in the 90s97 x 94

• Find out how far each number is from 100
• The 1st two numbers equal the sum of the
  differences subtracted from 100
• The last two numbers equal the
  product of the differences

100-(36)
3(6)
91 18
20
Multiplying Two Numbers Near 100109 x 106

• First Number is always 1
• The middle two numbers
  the sum on the units digits
• The last two digits the
  product of the units digits

1
96
9(6)
1 15 54
21
Multiplying Two Numbers With 1st Numbers And A
0 In The Middle402 x 405

• The 1st two numbers the product of the hundreds
  digits
• The middle two numbers the sum of the
  units x the hundreds digit
• The last two digits the product of the units
  digits

4(4)
4(25)
2(5)
16 28 10
22
Multiplying By 336718 x 3367
10101 Rule

• Divide the non-3367 by 3
• Multiply by 10101

18/3 6 x 10101
60606
23
Multiplying A 2-Digit By 1192 x 11
121 Pattern
(ALWAYS WORK FROM RIGHT TO LEFT)

• Last digit is the units digit
• The middle digit is the sum of the tens and the
  units digits
• The first digit is the tens digit any carry

91
92
2
10 1 2
24
Multiplying A 3-Digit By 11192 x 11
1221 Pattern
(ALWAYS WORK FROM RIGHT TO LEFT)

• Last digit is the units digit
• The next digit is the sum of the tens and the
  units digits
• The next digit is the sum of the tens and the
  hundreds digit carry
• The first digit is the hundreds digit any carry

191
92
2
11
2 1 1 2
25
Multiplying A 3-Digit By 111192 x 111
12321 Pattern
(ALWAYS WORK FROM RIGHT TO LEFT)

• Last digit is the units digit
• The next digit is the sum of the tens and the
  units digits
• The next digit is the sum of the units, tens and
  the hundreds digit carry
• The next digit is the sum of the tens and
  hundreds digits carry
• The next digit is the hundreds digit carry

191
92
2
11
1921
2 1 3 1 2
26
Multiplying A 2-Digit By 11141 x 111
1221 Pattern
(ALWAYS WORK FROM RIGHT TO LEFT)

1. Last digit is the units digit
2. The next digit is the sum of the tens and the
   units digits
3. The next digit is the sum of the tens and the
   units digits carry
4. The next digit is the tens digit carry

4
41
1
41
4 5 5 1
27
Multiplying A 2-Digit By 10193 x 101

1. The first two digits are the 2-digit number x1
2. The last two digits are the 2-digit number x1

93(1)
93(1)
93 93
28
Multiplying A 3-Digit By 101934 x 101

1. The last two digits are the last two digits
   of the 3-digit number
2. The first three numbers are the 3-digit
   number plus the hundreds digit

9349
34
943 34
29
Multiplying A 2-Digit By 100187 x 1001

1. The first 2 digits are the 2-digit number x 1
2. The middle digit is always 0
3. The last two digits are the 2-digit number x 1

87(1)
0
87(1)
87 0 87
30
Halving And Doubling52 x 13

1. Take half of one number
2. Double the other number
3. Multiply together

52/2
13(2)
26(26) 676
31
One Number in the Hundreds And One Number In The
90s95 x 108

1. Find how far each number is from 100
2. The last two numbers are the product of
   the
   differences subtracted from 100
3. The first numbers the small number difference
   from 100 increased by
   1 and subtracted from the larger number

108-(581)
100-(5x8)
94 60
32
Fraction Foil (Type 1)8 ½ x 6 ¼

1. Multiply the fractions together
2. Multiply the outside two number
3. Multiply the inside two numbers
4. Add the results and then add to the
   product of the whole numbers

(8)(6)1/2(6)1/4(8)
(1/2x1/4)
53 1/8
33
Fraction Foil (Type 2)7 ½ x 5 ½

1. Multiply the fractions together
2. Add the whole numbers and
   divide by the
   denominator
3. Multiply the whole numbers and
   add to previous step

(75)6
(1/2x1/2)
18 1/4
34
Fraction Foil (Type 3)7 ¼ x 7 ¾

1. Multiply the fractions together
2. Multiply the whole number by
   one more than the whole number

(7)(71)
(1/4x3/4)
56 3/16
35
Adding Reciprocals7/8 8/7

• Keep the denominator
• The numerator is the difference of
  the two numbers squared
• The whole number is always two plus
  any carry from the fraction.

36
Percent Missing the Of36 is 9 of __

• Divide the first number by
  the percent number
• Add 2 zeros or move the
  decimal two places to the right

36/9
00

400
37
Base N to Base 10 Of426 ____10

• Multiply the left digit times the base
• Add the number in the units column

4(6)2

2610
38
Multiplying in Bases4 x 536___6

• Multiply the units digit by the multiplier
• If number cannot be written in base n subtract
  base n until the digit
  can be written
• Continue until you have the answer

4x312 subtract 12 Write 0

4x520222 subtract 18 Write 4
Write 3
3406
39
N/40 to a or Decimal21/40___decimal

• Mentally take off the zero
• Divide the numerator by the denominator
  and write down the digit
• Put the remainder over the 4 and write the
  decimal without the decimal point
• Put the decimal point in front of the numbers

5
25
.
21/4
1/4
40
Remainder When Dividing By 9867/9___remainder

• Add the digits until you get a single digit
• Write the remainder

86721213

3
41
Base 8 to Base 27328 ____2
421 Method

• Mentally put 421 over each number
• Figure out how each base number
  can be written with a 4, 2 and 1
• Write the three digit number down

421
421
421
7
3
2
111
011
010
42
Base 2 to Base 8 Of1110110102 ___8
421 Method

1. Separate the number into groups
   of 3 from the right.
2. Mentally put 421 over each group
3. Add the digits together and write the sum

421
421
421
011
010
111
7
3
2
43
Cubic Feet to Cubic Yards3ft x 6ft x 12ft__yds3

• Try to eliminate three 3s by division
• Multiply out the remaining numbers
• Place them over any remaining 3s

3
12
6
3
3
3
1 x 2 x 4 4
Cubic yards
44
Cubic Feet to Cubic Yards44 ft/sec __mph

• Use 15 mph 22 ft/sec
• Find the correct multiple
• Multiply the other number

22x244
15x230 mph
Cubic yards
45
Subset ProblemsF,R,O,N,T______
SUBSETS

• Subsets2n
• Improper subsets always 1
• Proper subsets 2n - 1
• Power sets subsets

46
Repeating Decimals to Fractions.18___fraction
___

• The numerator is the number
• Read the number backwards. If a number has a
  line over it then there is a 9 in the denominator
• Write the fraction and reduce

18
2

99
11
47
Repeating Decimals to Fractions.18___fraction
_

• The numerator is the number minus
  the part that does not repeat
• For the denominator read the number backwards.
  If it has a line over it,
  it is a 9. if not it is a o.

18-1
17

90
90
48
Gallons Cubic Inches2 gallons__in3
(Factors of 231 are 3, 7, 11)

• Use the fact 1 gal 231 in3
• Find the multiple or the factor and adjust the
  other number. (This is a direct variation)

2(231) 462 in3
49
Finding Pentagonal Numbers5th Pentagonal __

• Use the house method)
• Find the square , find the triangular ,
  then add them together

1234 10
251035
25
5
5
50
Finding Triangular Numbers6th Triangular __

• Use the n(n1)/2 method
• Take the number of the term that you are looking
  for and multiply it by one more than that term.
• Divide by 2 (Divide before multiplying)

6(61)42
42/221
51
Pi To An Odd Power13____approximation

• Pi to the 1st 3 (approx) Write a 3
• Add a zero for each odd power
  of Pi after the first

3000000
52
Pi To An Even Power12____approximation

• Pi to the 2nd 95 (approx) Write a 95
• Add a zero for each even power
  of Pi after the 4th

950000
53
The More Problem17/15 x 17

1. The answer has to be more than the whole number.
2. The denominator remains the same.
3. The numerator is the difference in the two
   numbers squared.
4. The whole number is the original whole number
   plus the difference

(17-15)2
172
15
19 4/15
54
The Less Problem15/17 x 15

1. The answer has to be less than the whole number.
2. The denominator remains the same.
3. The numerator is the difference in the two
   numbers squared.
4. The whole number is the original whole number
   minus the difference

(17-15)2
15-2
17
13 4/15
55
Multiplying Two Numbers Near 1000994 x 998

• Find out how far each number is from 1000
• The 1st two numbers equal the sum of the
  differences subtracted from 1000
• The last two numbers equal the product of the
  differences written as a 3-digit number

1000-(62)
6(2)
992 012
56
The (Reciprocal) Work Problem1/6 1/5 1/X
Two Things Helping

1. Use the formula ab/ab.
2. The numerator is the product of the two numbers.
3. The deniminator is the sum of the two numbers.
4. Reduce if necessary

6(5)
65
30/11
57
The (Reciprocal) Work Problem1/6 - 1/8 1/X
Two Things working Against Each Other

1. Use the formula ab/b-a.
2. The numerator is the product of the two numbers.
3. The denominator is the difference of the two
   numbers from
   right to left.
4. Reduce if necessary

6(8)
8-6
24
58
The Inverse Variation Problem30 of 12 20
of ___

• Compare the similar terms as a reduced ratio
• Multiple the other term by the reduced ratio.
• Write the answer

59
Sum of Consecutive Integers123..20

• Use formula n(n1)/2
• Divide even number by 2
• Multiply by the other number

(20)(21)/2
10(21) 210
60
Sum of Consecutive Even Integers246..20

• Use formula n(n2)/4
• Divide the multiple of 4 by 4
• Multiply by the other number

(20)(22)/4
5(22) 110
61
Sum of Consecutive Odd Integers135..19

• Use formula ((n1)/2)2
• Add the last number and the first number
• Divide by 2
• Square the result

(191)/2
102 100
62
Finding Hexagonal NumbersFind the 5th
Hexagonal Number

1. Use formula 2n2-2n
2. Square the number and multiply by2
3. Subtract the number wanted from the previous
   answer

2(5)2 50
50-5
45
63
Cube PropertiesFind the Surface Area of a Cube
Given the space Diagonal of 12

1. Use formula Area 2D2
2. Cancel the 2 if possible
3. Multiply the remaing numbers

2(12)(12)/2
6(12)
72
64
Cube Properties
Find S, Then Use It To Find Volume or Surface
Area
65
Finding Slope From An Equation3X2Y10

• Solve the equation for Y
• The number in fron of X is the Slope

3X2Y10

Slope -3/2
66
Hidden Pythagorean Theorem Find The Distance
Between These Points(6,2) and (9,6)

• Find the distance between the Xs
• Find the distance between the Ys
• Look for a Pythagorean triple
• If not there, use the Pythagorean Theorem

9-63
6-24
3 4 5

5 12 13
3
4
5
7 24 25
8 15 17
The distance is 5
Common Pythagorean triples
67
Finding Diagonals Find The Number Of
Diagonals In An Octagon

• Use the formula n(n-3)/2
• N is the number of vertices in the polygon

8(8-3)/2

20
68
Finding the total number of factors 24 ________

• Put the number into prime factorization
• Add 1 to each exponent
• Multiply the numbers together

31 x 23

112 314
2x48
69
Estimating a 4-digit square root
7549 _______

• The answer is between 802 and 902
• Find 852
• The answer is between 85 and 90
• Guess any number in that range

8026400

8527225
87
9028100
70
Estimating a 5-digit square root
37485 _______

• Use only the first three numbers
• Find perfect squares on either side
• Add a zero to each number
• Guess any number in that range

192361

190-200
195
202400
71
C F
55C _______F

1. Use the formula F 9/5 C 32
2. Plug in the F number
3. Solve for the answer

9/5(55) 32

9932
131
72
C F
50F _______C

1. Use the formula C 5/9 (F-32)
2. Plug in the C number
3. Solve for the answer

5/9(50-32)

5/9(18)
10
73
2-Digit Number Times 1001
87 x 1001

1. 87 times 1
2. Put a zero in the middle
3. 87 x 1

87 0 87

87087
74
Finding The Product of the Roots
4X2 5X 6
a
b
c

1. Use the formula c/a
2. Substitute in the coefficients
3. Find answer

6 / 4 3/2

75
Finding The Sum of the Roots
4X2 5X 6
a
b
c

1. Use the formula -b/2a
2. Substitute in the coefficients
3. Find answer

-5 / 8

76
Estimation
999999 Rule
142857 x 26

1. Divide 26 by 7 to get the first digit
2. Take the remainder and add a zero
3. Divide by 7 again to get the next number
4. Find the number in 142857 and copy in a circle

26/7 3r6

6060/78
3 857142
77
Area of a Square Given the Diagonal
Find the area of a square with a diagonal of 12

1. Use the formula Area ½ D1D2
2. Since both diagonals are equal
3. Area ½ 12 x 12
4. Find answer

½ D1 D2

½ x 12 x 12
72
78
Estimation of a 3 x 3 Multiplication
346 x 291

1. Take off the last digit for each number
2. Round to multiply easier
3. Add two zeroes
4. Write answer

35 x 30

1050 00
105000
79
Adding Reciprocals
7 8

8 7

1. Work backwards
2. The fraction in the difference of the two number
   squared then put over the product of the numbers
   
3. Write the fraction
4. The whole number is always 2 plus any carry

8-71 121
2 1/56
80
Dividing by 11 and finding the remainder
7258 / 11_____

Remainder

1. Start with the units digit and add up every other
   number
2. Do the same with the other numbers
3. Subtract the two numbers
4. If the answer is a negative or a number greater
   than 11 add or subtract 11 until you get a number
   from 0-10

8210 75 12
10-12 -2 11 9
81
Multiply By Rounding
2994 x 6

• Round 2994 up to 3000
• Think 3000 x 6
• Write 179. then find the last two numbers by
  multiplying what you added by 6 and subtracting
  it from 100.

3000(6)179_ _

6(6)36 100-3664
17964
82
The Sum of Squares
(factor of 2)
122 242

• Since 12 goes into 24 twice
• Square 12 and multiply by 10

122144
144x10

1440
83
The Sum of Squares
(factor of 3)
122 362

• Since 12 goes into 36 three times
• Square 12 and multiply by 10
• Then divide by 2

122144
144x10

1440
720
84
The Difference of Squares
(Sum x the Difference)
322 - 302

• Find the sum of the bases
• Find the difference of the bases
• Multiply them together

323062

32-302
62 x 2 124
85
Addition by Rounding
(Sum x the Difference)
2989 456

• Round 2989 to 3000
• Subtract the same amount to 456, 456-11 445
• Add them together

298911 3000

456-11445
30004453445
86
123x9 A Constant
(1111Problem)
123 x 9 4

• The answer should be all 1s. There should be 1
  more 1 than the length of the 123 pattern.
• You must check the last number. Multiply the last
  number in the 123 pattern and add the constant.
  

3x9 4 31

1111
87
Supplement and Complement

Find The Difference Of The Supplement And The
Complement Of An Angle Of 40.

• The answer is always 90

90
88
Supplement and Complement

Find The Sum Of The Supplement And The Complement
Of An Angle Of 40.

• Use the formula 270-twice the angle
• Multiple the angle by 2
• Subtract from 270

270-80
190
89
Larger or Smaller
55
52

• Find the cross products
• The larger fraction is below the larger number
• The smaller number is below the smaller number

Larger 5/4

Smaller 13/11
90
Two Step Equations(Birthday Present Problem)

• Start with the answer and undo the
  operations using reverse order of operations

11112

12 x3 36
91
Relatively Prime(No common Factors Problem)
One is relatively prime to all numbers

How Many s less than 20 are relatively prime to
20?

• Put the number into prime factorization
• Subtract 1 from each exponent and multiply
  out all parts separately
• Subtract 1 from each base
• Multiply all parts together

2 x 1 x 1 x 4 8
92
Product of LCM and GCF

Find the Product of the GCF and the LCM of 6 and
15

• Multiple the two numbers together

6 x 15 90

93
Estimation

15 x 17 x 19

• Take the number in the middle and cube it

1734913

94
Sequences-Finding the Pattern

7, 2, 5, 8, 3, 14
Find the next number in this pattern

• If the pattern is not obvious try looking at
  every other number. There may be two patterns
  put together

7, 2, 5, 8, 3, 14

1
95
Sequences-Finding the Pattern

1, 4, 5, 9, 14, 23
Find the next number in this pattern

• If nothing else works look for a Fibonacci
  Sequence where the next term is the sum of the
  previous two

1, 4, 5, 9, 14, 23

142337
96
Degrees Radians

900 _____
Radians

• If you want radians use Pi X/180
• If you want degrees use 180 x/Pi

90(Pi)/180

Pi/2